Simplify; express your answer in exponential form. Assume $z\neq 0, x\neq 0$. $\dfrac{{(z^{-4}x^{-3})^{-1}}}{{(z^{3}x^{5})^{2}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{-4}x^{-3})^{-1} = (z^{-4})^{-1}(x^{-3})^{-1}}$ On the left, we have ${z^{-4}}$ to the exponent ${-1}$ . Now ${-4 \times -1 = 4}$ , so ${(z^{-4})^{-1} = z^{4}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{-4}x^{-3})^{-1}}}{{(z^{3}x^{5})^{2}}} = \dfrac{{z^{4}x^{3}}}{{z^{6}x^{10}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{4}x^{3}}}{{z^{6}x^{10}}} = \dfrac{{z^{4}}}{{z^{6}}} \cdot \dfrac{{x^{3}}}{{x^{10}}} = z^{{4} - {6}} \cdot x^{{3} - {10}} = z^{-2}x^{-7}$